Assume that you have developed a good set of tools allowing you to decide which real functions of one real variable, $f\colon\mathbb{R}\to\mathbb{R}$, are continuous. (or, even more general, what are the continuity points of $f$).

(Q): How can such a tool-box be utilized to decide on the continuity of the functions $g\colon\mathbb{R}^n\to\mathbb{R}$ of $n$ real variables?

This is one of the questions which must be faced by any student taking multivariable calculus. Of course, such a student is following the footsteps of many generations of mathematicians, which were, and still are, struggling with the same general question. The aim of this article is to present the history and the current research related to this subject in a real analysis perspective, rather than in a more general, topological perspective.

In addition to surveying the results published so far, this exposition also includes several original results, as well as some new simplified versions of the proofs of older results. We also recall several intriguing open problems.

Based on the Zermelo-Fraenkel system of axioms ZF, we introduce a theory of essential closures. It is a generalization of the concept of topological closures. A typical essential closure collects all points which are essential with respect to a submeasure; hence it is called a submeasure closure. One of our main results states that a “nice” essential closure must be a submeasure closure. Many examples of known and new submeasure closures are discussed and their applications are demonstrated, especially in the study of the supports of measures.

Up to a set of measure zero we characterize the sets of non-differentiability of functions with everywhere finite lower scaled oscillation.

We present various inequalities for means in two variables. One of our results states that the inequalities $$ 0\leq \frac{1}{M_r} -\frac{1}{M_s} \leq \frac{1}{G}-\frac {1}{A } \quad{(r,s\geq 0)} $$ hold for all $x,y>0$ if and only if $0\leq s-r\leq 1$. Here, $A=A(x,y)=(x+y)/2$, $G=G(x,y)=\sqrt{xy}$ and $M_t=M_t(x,y)=[(x^t+y^t)/2]^{1/t}$ denote the arithmetic, geometric and power mean of $x$ and $y$, respectively.

Answering one of the real function problems suggested by A. Maliszewski, the existence of a bounded Darboux function of the Sierpiński first class which cannot be expressed as a difference of two bounded lower semicontinuous functions is proved. As the reply to the other Maliszewski question, we show there exists an almost everywhere continuous Darboux function of the Sierpiński first class which is not a difference of two almost everywhere continuous lower semicontinuous functions.

The definition of the $L_{p}$-derivative is such that it involves only the absolute value of the function and therefore the definition of $L_{p}$-derivates is not possible from the definition of $L_{p}$-derivative. Therefore, a special technique is used to define them and relations between $L_{p}$-derivates and Peano, approximate Peano and Borel derivates are studied.

Sufficient conditions for the convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in the classes $L_p$, $p>1$, are obtained in the case where rectangular partial sums $S_n(x;f)$ of this series have numbers $n=(n_1,\dots,n_N)\in \mathbb Z^N$, $N\geq 3$, such that of $N$ components only $k$ ($1\leq k\leq N-2$) are elements of some lacunary sequences. Earlier, in the case where $N-1$ components of the number $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained for functions in the classes $L_p$, $p>1$, by M. Kojima (1977), and for functions in Orlizc classes by D. K. Sanadze, Sh. V. Kheladze (1977) and N. Yu. Antonov (2014). Note that presence of two or more “free” components in the number $n$, as follows from the results by Ch. Fefferman (1971) and M. Kojima (1977), does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\geq 3$ even in the class of continuous functions.

The study of functions of generalized bounded variation (VBG) and generalized absolute continuity (ACG) that appears in Saks’s treatise Theory of the Integral can be thoroughly reworked by using some aspects of the theory of variational measures proposed originally by Ralph Henstock and extended by many others. We present a development of these concepts and use it for a characterization of the Denjoy-Khintchine integral.

There is a function $F:[\mathfrak{c}]^{\lt\omega}\to\{0,1\}$ such that if $A\subseteq [\mathfrak{c}]^{\lt\omega}$ is uncountable, then $\{F(a\cup b):a,b\in A, a\neq b\}=\{0,1\}$. A corollary is that there is a function $f:\mathbb{R}\to\{0,1\}$ such that if $A\subseteq\mathbb{R}$ is uncountable, $2\leq k\lt\omega$, then both 0 and 1 occur as the value of $f$ at the sum of $k$ distinct elements of $A$. This was originally proved by Hindman, Leader, and Strauss under CH, and they asked if it holds in general.

Under Martin’s Axiom, it is proved that there exists an absolute null subset of the Euclidean plane $\mathbb{R}^2$, the orthogonal projections of which on all straight lines in $\mathbb{R}^2$ are absolutely nonmeasurable. A similar but weaker result holds true within the framework of ZFC set theory.

We study some basic properties of Lorentz sequence spaces. Description of multiplication operators generated by a sequence is presented. We calculate the essential norm of multiplication operators acting on Lorentz sequence spaces.